Midterm1 soln

Midterm1 soln - MATH150 Introduction to Ordinary...

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1 MATH150 Introduction to Ordinary Differential Equations Midterm Exam Suggested Solution Prepared by CHAU Suk Ling, Pat 1. Find the solution () yy x = of the following initial value problem: '( ) x x yy e e =− , (0) 2 y = − . Solution 2 2 xx dy ye e dx ydy e e dx y ee C =+ + Putting 0 x = and 2 y , we have 22 0 C C =+ = Thus 2 2 2( ) x x y ee e + Note that we have to take the negative square root as (0) 2 y = − . 2. (a) With λ and b given, find the solution x xt = of the following initial value problem. t x xb e += ± ; (0) 0 x = . (b) Determine the value of t at which () x t is maximum. Assume 0 > and 0 b > . Solution (a) An integrating factor dt t = = . So tt t x eb e e d t bdt bt C λλ = = Putting 0 t = and 0 x = , we have 0 C = . Thus t t x t x bte = = (b) Since '( ) (1 ) t x tb e b t e t b e −− = , 0 = when 1 t = .
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2 Since b and t e λ are positive, when 1 t < , '( ) 0 xt > and so () x t is increasing; when 1 t > , 0 < x t is decreasing. This shows that () x t attains global maximum at 1 t = .
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Midterm1 soln - MATH150 Introduction to Ordinary...

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